Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

• Published in: Mathematics (Basel) 2023-01, Vol.11 (1), p.46
• Main Authors: Melaibari, Ammar, Mohamed, Salwa A., Assie, Amr E., Shanab, Rabab A., Eltaher, Mohamed A.
• Format: Article
• Language: English
• Subjects: 2D-DIQM; 2D-FG Pate; Behavior; Boundary conditions; Composite materials; Deformation; Distribution functions; elastic foundation; Elasticity; Equations of motion; Foundations; Free vibration; Functionally gradient materials; Hamilton's principle; Kinematics; Mathematical analysis; Mechanical properties; Optimization techniques; Parametric analysis; Partial differential equations; Plates (Engineering); Porosity; Porous materials; Porous plates; Quadratures; Resonant frequencies; Shear strain; Stress analysis; Topology; Trigonometric functions; unified plate theories; Vibration; Vibration research; Viscoelasticity
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11010046

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.